Unsigned distance functions offer a powerful and flexible implicit surface representation that, unlike their signed counterparts, allow for surfaces that are open, non-orientable, or non-manifold. We consider the problem of reconstructing arbitrary surfaces from a finite set of samples of unsigned distance data. Existing methods for mesh reconstruction from distance data rely on sign information, accurate gradients, a corresponding continuous distance function, or extensive data-dependent training. However, they fail when applied to input that is both discrete and unsigned. Inspired by this challenge, we study the power diagram generated by the distance samples and propose a novel theoretical concept, the superpower contour, which we prove converges to the true surface in the limit of sampling density. We use this superpower contour as an initial surface proxy and design an algorithm that leverages it to produce a polygonal mesh approximating the unknown true geometry. Our method vastly outperforms other conceivable strategies for the discrete unsigned distance reconstruction task, and sets the stage for future work on this mathematically rich problem.

翻译：无符号距离函数提供了一种强大且灵活的隐式曲面表示方法，与有符号距离函数不同，它能够表示开放、不可定向或非流形曲面。本文研究从有限的无符号距离数据样本中重建任意曲面的问题。现有基于距离数据的网格重建方法依赖于符号信息、精确梯度、对应的连续距离函数或大量数据驱动的训练。然而，当输入为离散且无符号数据时，这些方法均失效。受此挑战启发，我们研究了距离样本生成的幂图，并提出了一种新颖的理论概念——超动力等值面（superpower contour），并证明其在采样密度趋于无穷时收敛于真实曲面。我们将该超动力等值面作为初始曲面代理，并设计了一种利用该代理生成逼近未知真实几何的多边形网格的算法。我们的方法在离散无符号距离重建任务中显著优于其他可行策略，并为这一数学内涵丰富问题的未来研究奠定了基础。